Micro-resonator design implementing internal resonance for MEMS applications

ABSTRACT

Frequency stabilization is provided in a microelectromechanical systems (MEMS) oscillator via tunable internal resonance (IR). A device comprises a MEMS resonator comprising a stepped-beam structure that is a thin-layer structure. The resonator may be configured to implement IR. The stepped-beam structure may be configured to provide flexibility to adjust modal frequencies into a n:m ratio, wherein n and m are integers. The thin-layer structure provides frequency tunability by controlling the mid-plane stretching effect with an applied DC bias. The thin-layer structure compensates for a frequency mismatch from a n:m ratio due to a fabrication error. The MEMS resonator may be an oscillator.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a national stage application filed under 35 U.S.C. §371 of PCT/US2020/022674 filed Mar. 13, 2020, which claims the benefitof U.S. provisional patent application No. 62/818,234, filed on Mar. 14,2019, and entitled “MICRO-RESONATOR DESIGN IMPLEMENTING INTERNALRESONANCE FOR MEMS APPLICATIONS,” the disclosure of which is expresslyincorporated herein by reference in its entirety.

STATEMENT OF GOVERNMENT SUPPORT

This invention was made with government support under D16AP00110 awardedby the DARPA (Defense Advanced Research Projects Agency). The governmenthas certain rights in the invention.

BACKGROUND

Microelectromechanical systems (MEMS) attract great interest due totheir compact dimensions, fast response, high sensitivity, and low powerconsumption, and are widely used in many industrial, commercial, andscientific applications. Micromechanical resonators are one category ofMEMS that are designed to operate at or near their resonant frequencies.These resonators are often fabricated into beam or disk structures. Theultralow damping combined with low effective mass allows them to operateat very high resonant frequencies with high Q-factors. In addition, theyhave diverse actuation/detection mechanisms. These attributes exploitmicromechanical resonators into a key element in micromechanicaloscillators and sensors for providing reference frequencies and highsensitivity.

One of the most important attributes of resonators is the frequencystability, which determines their performance of sensitivity andreliability. However, when their dimensions shrink to micro- and evennano-scale, the frequency can fluctuate due to various noise sourceseven with tiny energy such as thermal noise, absorbing/desorbingmolecules, and additive noises from actuation and transduction circuits.In other words, instabilities that are negligible in macro-scale devicesbecome dominated when the dimensions of the oscillators shrink down tothe micro- and nano-scale. Temperature fluctuations, moisture change,adsorbing/desorbing molecules, even fluctuations in the number ofphotons can all affect the frequency stability. Additionally,micromechanical oscillators often fall in the nonlinear regime when itis strongly driven to acquire the large signal-to-noise ratio. Anundesired consequence of nonlinear operation is that extensive frequencyfluctuations are induced to the oscillator which further degrades theirperformances. As timing devices, the key role of oscillator is toprovide stable reference frequencies, but all of these instabilitiesprevent the advances of micromechanical oscillators.

Various strategies for frequency stabilization have been proposed inprevious studies, such as achieve ultra-high Q factor andsynchronization of oscillators. However, no published frequencystability measurements attain the limit set by the thermo-mechanicalnoise, and are still several orders higher than the thermo-mechanicalnoise limit.

It is known that the mechanism of internal resonance (IR) improves thefrequency stability in MEMS oscillators and MEMS sensors. IR, however,is not easy to realize in a prismatic structure, because it can be onlytriggered when a commensurate condition between two (or more) involvedmodes is satisfied. It is with respect to these and other considerationsthat the various aspects and embodiments of the present disclosure arepresented.

SUMMARY

Certain aspects of the present disclosure relate to frequencystabilization in a microelectromechanical systems (MEMS) oscillator viatunable internal resonance (IR). Some non-limiting examples ofembodiments of the present disclosure include the following.

An implementation comprises a MEMS non-prismatic resonator, with astepped-beam structure that is a thin-layer structure. The resonator maybe configured to implement IR. The stepped-beam structure may beconfigured to provide flexibility to adjust modal frequencies into a n:mratio, wherein m and n are integers, and m and n can be the same integeror can be different integers from each other depending on theimplementation. The thin-layer structure provides frequency tunabilityby controlling the mid-plane stretching effect with an applied DC bias.The thin-layer structure compensates for a frequency mismatch from a n:mratio due to a fabrication error. The MEMS resonator may be anoscillator.

According to some aspects, a mistuning between two flexural modes of theMEMS resonator can be precisely controlled by tuning a DC bias and,through strong coupling between the two flexural modes, a broader rangeof frequency stabilization is achieved by IR.

In some implementations, the device may comprise a steppedclamped-clamped silicon microbeam. The stepped clamped-clamped siliconmicrobeam may be fabricated by a MEMS fabrication flow to implement anIR mechanism.

In some aspects, the present invention relates to systems and techniquesfor a thin-layer stepped-beam MEMS resonator that can readily implementTR. A thin-layer stepped-beam structure allows for the attainment of thecommensurate condition between two vibrational modes. Moreover, highfrequency tunability achieved by its thin structure enables tuning theIR to its optimal condition by adjusting an applied DC voltage.

This summary is provided to introduce a selection of concepts in asimplified form that are further described below in the detaileddescription. This summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing summary, as well as the following detailed description ofillustrative embodiments, is better understood when read in conjunctionwith the appended drawings. For the purpose of illustrating theembodiments, there is shown in the drawings example constructions of theembodiments; however, the embodiments are not limited to the specificmethods and instrumentalities disclosed. In the drawings:

FIG. 1 is an illustration of a modal analysis used to predict dimensionsof resonators which can satisfy commensurability between second andthird modes;

FIG. 2 is a scanning electron microscopy (SEM) image of animplementation of a batch-fabricated thin-layer stepped-beam resonator;

FIG. 3 is a diagram of a close-loop configuration system withelectrostatic actuation and optical detection;

FIG. 4 shows mode frequencies of first three flexural modes underdifferent DC bias voltages;

FIG. 5 shows the ratio of third to second flexural mode frequenciesunder different DC bias voltages;

FIG. 6 is an illustration of another example micromechanical resonatorand electrodes;

FIGS. 7 and 8 show the finite element analysis (FEA) predicted modeshape of second mode with mode frequency of 1835 kHz, and predicted modeshape of third mode with mode frequency of 3671 kHz, respectively;

FIG. 9 is a schematic diagram of an implementation of an RF/LO mixingmeasurement setup;

FIGS. 10 and 11 show illustrations of the comparison of the harmoniccontent in FFT-based spectral responses;

FIGS. 12, 13, and 14 show the experimental characterization of thenonlinear IR frequency responses; and

FIGS. 15 and 16 respectively show that first three flexural mode shapeswith a symmetrical configuration and asymmetrical configuration.

DETAILED DESCRIPTION

This description provides examples not intended to limit the scope ofthe appended claims. The figures generally indicate the features of theexamples, where it is understood and appreciated that like referencenumerals are used to refer to like elements. Reference in thespecification to “one embodiment” or “an embodiment” or “an exampleembodiment” means that a particular feature, structure, orcharacteristic described is included in at least one embodimentdescribed herein and does not imply that the feature, structure, orcharacteristic is present in all embodiments described herein.

A non-prismatic (e.g., with varying dimension) micromechanical resonatoror nanomechanical resonator that can readily implement internalresonance (IR) is described. The non-prismatic resonator may be astepped-beam resonator in some implementations. IR can happen in anonlinear system, when frequencies of two or more resonant modes arecommensurable or nearly commensurable (e.g., n:m ratio, wherein m and nare integers, and m and n can be the same integer or can be differentintegers from each other depending on the implementation). When the IRhappens, the engaged modes in the commensurable condition caneffectively exchange energy internally. Such an “internal” energytransfer happens much faster than the “external” energy transferhappening between the structure and the environment (e.g., energydissipation from the system to the environment, energy pumping from thedriving force to the system).

Frequency stability is a key parameter to determine the performance of amicroelectromechanical (MEM) resonator. When a MEM resonator is drivenby a single-frequency actuation and conditions for IR are satisfied, twoengaged modes are simultaneously resonated with their phases lockedtogether through a strong and effective intermodal energy transfer. Theincreased inertia of these two active resonant modes leads to thefrequency stabilization in both frequency outputs. In an open-loopexperimental setting, the IR achieved a six-fold improvement in thefrequency stability.

There are many advantages from using the mechanism of IR, as implementedin the resonator designs provided herein. For example, (1) frequencystabilization in MEMS oscillators and nanoelectromechanical systems(NEMS) oscillators, (2) strategies to control energy transfer inmicro-resonators and nano-resonators, (3) multiple modes for multi-modalsensing capability, and (4) frequency control in radio-frequency (RF)MEMS.

Described herein is a (i) non-prismatic, (ii) clamped-clamped, and/or(iii) thin layer of beam (or other 2D structural) design as a platformthat can readily implement IR. IR is not easy to realize in a prismatic(constant dimension) structure because a clamped-clamped prismaticstructure typically does not provide a n:m ratio. A stepped-beamstructure provides ample flexibility to adjust modal frequencies into an:m ratio, wherein m and n are integers during the design process.Internal coupling can be significantly enhanced when the system has anasymmetric resonant mode shape. Coupling the second and third modes ismore effective than coupling other modes. The clamped-clamped structureinherently provides a nonlinear coupling between mode via mid-planestretching. The mode frequency of a thin structure is easy to tune byusing various mechanisms (e.g., tuning mid-plane stretchingelectrostatically by applying DC bias).

As described further herein, 1:2 IR is implemented on the second andthird flexural modes. Through this strong coupling, a broad IR range isachieved and frequency stability is improved close to itsthermo-mechanical noise limit because IR stabilizes frequency from themechanical domain.

FIG. 1 is an illustration of a modal analysis 100 used to predictdimensions of resonators which can satisfy commensurability betweensecond and third modes. The second and third mode frequencies 110, 120are 236.49 kHz and 473.75 kHz, respectively. In other words, as shown inFIG. 1 , the modal analysis prediction of second and third flexuralmodes are 236.49 kHz and 473.75 kHz, respectively, which satisfies a 1:2commensurate condition. Thin-layer structures are more sensitive tomid-plane stretching which is determined by DC bias, and thus have hightunability to compensate the fabrication variance. A clamped-clampedbeam structure may also be used, and has higher resonant frequenciescomparing with other structures with similar dimensions.

FIG. 2 is a scanning electron microscopy (SEM) image 200 of animplementation of a batch-fabricated thin-layer stepped-beam resonator210. The resonator 210 may be fabricated by conventionalmicrofabrication process. The scale bar is 100 μm. A thin-layerstepped-beam micromechanical resonator 210 is provided to couple theirsecond and third flexural modes by 1:2 IR. Prismatic structures havetheir own eigenvalue ratio between mode frequencies, which are not oftencommensurate. To this end, the designed structure comprises a wider partof L₁=187 μm, W₁=8.9 μm and a narrow part of L₂=68 μm, W₂=4 μm with auniform thickness T of 500 nm.

A MEMS resonator, such as the resonator 210, may be fabricated byconventional flow. Briefly, a microbeam was patterned in asilicon-on-insulator (SOI) wafer with 500 nm device layer and 2 μm boxlayer using photolithography technique. Then, reactive ion etching (RIE)is employed to etch exposed area and then the wafer is immersed inhydrofluoric acid solution to release the beam. The device is mounted ina vacuum chamber, with pressure around 3 mTorr. An AC signal produced byan internal function generator of a lock-in amplifier (Zurich InstrumentHF2LI) may be applied to drive the microbeam. The involved vibrationalamplitudes may be detected optically by a laser Doppler Vibrometer (LDV,Polytec OFV-534 sensor & OFV-5000 controller), since they areout-of-plane flexural modes. The LDV output a voltage signal which isproportional to displacement of microbeam. This resulting signal isinduced to the lock-in amplifier and a spectrum analyzer (TektronixRSA603A) simultaneously, which not only measure higher or lowerharmonics but also detect the full spectrum of IR.

Micromechanical resonators can implement the IR. By controlling the DCbias, the commensurate relationship between second and third flexuralmodes can be achieved, which is the prerequisite of IR. Toexperimentally characterize IR and study frequency stability, set up aRF/LO mixing measurement system to avoid influence from the parasiticfeedthrough current and acquire unmasked motional signal. In thissetting, verify that the IR happens in the stepped-beam resonator andits frequency stability improves by six-fold. The mechanism of IRstabilizes the frequency from the mechanical domain, which is the mainsource of frequency fluctuation.

These batch-fabricated resonators were actuated electrostatically anddetected optically by a closed-loop experimental configuration as shownin FIG. 3 . FIG. 3 is a diagram of a close-loop configuration system 300with electrostatic actuation and optical detection for a resonator 305.By changing the DC bias voltage (V_(dc) 302), mode frequencies wereelectrostatically tuned to be commensurate to satisfy the required IRcondition. Optical detection by LDV 310 (controlled by an LDV controller315) measured tiny motions precisely and output a voltage proportionalto the motional displacement. Another advantage of LDV measurement isthat it avoids undesired electrical signal, such as parasiticfeedthrough signal. A phase shifter 320, phase-locked loop (PLL) 325,and digital frequency counter 330 may be used to study the frequencystability. The output of the PLL 325 is provided to a driving control335, which provides V_(ac) 337 to the resonator, and also providesoutput to the frequency counter 330.

Regarding a closed-loop characterization, the instrumental connection ofclosed-loop configuration is the same as the open-loop configuration,but a built-in phase shifter 320, PLL 325, and digital frequency counter330 of the lock-in amplifier are employed here. The output signal of theLDV 310 is phase shifted by the phase shifter 320 and then induced tothe PLL 325. By using a PID controller, the PLL 325 keeps phasedifference between input and output of a lock-in amplifier. The built-insignal generator is used to set the amplitude of output signal. Thus,the resulting signal with a constant amplitude and locked phase isproduced by the lock-in amplifier and used to excite the MEMS resonator305. The PLL 325 locks the oscillator at a desired point in thefrequency-to-amplitude curve. Then, the frequency fluctuation of thisresonator 305 can be detected by measuring the frequency of theoscillations with the digital frequency counter 330.

The environment of FIG. 3 is only one example of a suitable computingenvironment and is not intended to suggest any limitation as to thescope of use or functionality. The devices of FIG. 3 are well known inthe art and need not be discussed at length here. Numerous other generalpurpose or special purpose computing devices environments orconfigurations may be used. Examples of well known computing devices,environments, and/or configurations that may be suitable for useinclude, but are not limited to, personal computers, server computers,handheld or laptop devices, multiprocessor systems, microprocessor-basedsystems, network personal computers (PCs), minicomputers, mainframecomputers, embedded systems, distributed computing environments thatinclude any of the above systems or devices, and the like.

Computer-executable instructions, such as program modules, beingexecuted by a computer may be used. Generally, program modules includeroutines, programs, objects, components, data structures, etc. thatperform particular tasks or implement particular abstract data types.Distributed computing environments may be used where tasks are performedby remote processing devices that are linked through a communicationsnetwork or other data transmission medium. In a distributed computingenvironment, program modules and other data may be located in both localand remote computer storage media including memory storage devices.

To experimentally realize the IR, first tune the system to satisfy thecommensurate condition between second and third flexural modes. Themeasurements may be performed at room temperature and under vacuum(pressure less than 3 mTorr). FIG. 4 shows mode frequencies of firstthree flexural modes under different DC bias voltages. The modefrequencies were characterized from thermal-mechanical noise underdifferent DC bias. The inset show the thermal-mechanical noise when DCbias voltage was 19.1V.

Thus, the first three flexural mode frequencies were obtained bymeasuring the thermo-mechanical responses, as shown in the inset of FIG.4 , while a DC bias voltage was applied to the substrate to tune thefrequencies electrostatically. Because the DC bias influences themid-plane stretching of the microbeam structure, its mode frequencieswere tuned as shown in FIG. 4 . In a common electrostatically actuatedresonator system, mode frequencies decrease with the DC bias voltagebecause of the softening effect of electrostatic force. In this system,on the contrary, the mode frequencies increase with the DC bias voltage.The thickness of the resonator was just 500 nm, but the gap betweenresonator and substrate was 2 μm. This large gap-to-thickness ratioresulted in the mode frequencies increasing with DC bias voltage andimproved the tunability of the system.

FIG. 5 shows the ratio of third to second flexural mode frequenciesunder different DC bias voltages, when finely adjusted around the 1:2commensurate condition and eventually achieved exact 1:2 ratio when theDC bias was 19.1V. When DC bias voltage was around 19.1V, the ratio isclose to commensurate. Commensurability can be finely tuned by tuningthe DC bias voltage.

The prerequisite of IR is the commensurability and provided herein is aDC tuning strategy to satisfy commensurability within reasonablefabrication variance. Other tuning methods can be also applied in widerange of oscillators. For example, in a Duffing oscillator, resonantfrequency increases with driving amplitude (amplitude-frequency effect)because of hardening effect of nonlinearity. So, when two modes arenearly commensurate, use this amplitude-frequency effect to increase ordecrease driving force to satisfy commensurability. Another tuningmethod is electrothermally tuning. By applying a DC voltage between twoanchors of the clamped-clamped beam, a current passes through the beamand heats it. The stiffness of the beam is increased due to the Joulesheating effect and thus, the modal frequencies of the beam change.Combining these tuning methods with advanced fabrication techniques, IRcould be widely applied to micromechanical and nanomechanicaloscillators.

Thus, designed and fabricated herein is a micromechanical resonator withthin-layer, stepped-beam structure. Through DC bias tuning strategy, a1:2 commensurate relation is achieved between second and third flexuralmodes. Through this strong nonlinear coupling, unique M-shape resonantamplitude curves happens instead of simply hardening or softening curvesand a broad IR range is achieved. The frequency stability is improved bythis IR mechanism and after stabilization, the measured Allan Deviationis close to its thermal-mechanical noise limit.

Another implementation is described with respect to FIGS. 6-8 . FIG. 6is an illustration of an example micromechanical resonator 605 andelectrodes 610, 615, 620, 625. The resonator 605 and electrodes 610,615, 620, 625 are sealed within its own vacuum chamber protecting theresonator from environmental disturbances. The resonant beam of theresonator 605 vibrates in the flexural mode in the lateral direction.The stepped-beam with the dimensions of a wider part of L₁=172 μm, W₁=5μm and a narrower part of L₂=50 μm, W₂=3 μm with a uniform thickness of40 μm gives the second of third mode frequencies of 1835 kHz and 3671kHz, respectively, with their mode shapes shown in FIGS. 7 and 8 .

FIGS. 7 and 8 show the finite element analysis (FEA) predicted modeshape of second mode with mode frequency of 1835 kHz, and predicted modeshape of third mode with mode frequency of 3671 kHz, respectively.

To internally couple the second and third vibrational modes, thedimensions of a stepped-beam resonator were first determined to enforcea 1:2 ratio between these mode frequencies. A linear modal analysis wasconducted by a commercial finite element analysis (FEA) software COMSOLon a doubly clamped stepped-beam for various sets of width and length ofthe beam.

FIG. 9 is a schematic diagram of an implementation of an RF/LO mixingmeasurement setup 900 for use with a resonator 905 in accordance withthose described herein. A DC bias voltage from a DC power source 920 isapplied to a bias electrode of the resonator 905, via a bias-T 915,while an AC voltage is applied to an actuation electrode of theresonator 905 via a lock-in amplifier 925, resulting in an electrostaticforce to actuate the beam of the resonator 905 in the lateral direction.The dynamic motion of beam, in turn, introduces a time-varyingcapacitance between the beam and detection electrode. Finally, the ACcurrent including the motional signal is sensed from the outputelectrode by a transimpedance amplifier (TIA) 930 to convert and amplifythe current signal to a voltage signal.

In such a capacitive transduction, it is essential to eliminate theparasitic feedthrough capacitance that can mask the motional current. Assuch, use a mixing measurement system, so called RF/LO mixingmeasurement setup 900, as shown in FIG. 9 . The lock-in amplifier 925provides a sweeping RF signal to the actuation electrode of resonator905, while a local oscillator (LO) signal provided by a functiongenerator 910 is connected to the bias electrode of the resonator 905via bias-T 915. The mixing of RF and LO signals results in a sinusoidalforce with a frequency that equals to the frequency difference of RF andLO signals. When the frequency difference of RF and LO signals is set tobe around the mode frequency (i.e., ω_(RF)−ω_(LO)≈ω₀), this sinusoidalforce can be used to actuate the beam of the resonator 905. Through theTIA 930, the output current is converted to an amplified voltage signaland can be measured by either the spectrum analyzer 935 or the lock-inamplifier 925. The principle of this RF/LO mixing measurement is tocircumvent parasitic feedthrough signal, because the parasitic signalsexist on RF and LO frequencies rather than the frequency of drivingforce.

Here, the DC bias voltage can influence the mid-plane stretching of thebeam of the resonator and, thus, change its modal frequencies. By tuningthe DC bias, the IR range can be reached that satisfies the required 1:2frequency commensurate condition between second and third flexural modefrequencies. This DC-tuning strategy may be used to overcome fabricationvariances in devices.

The intermodal coupling of IR mechanism has a direct impact on frequencystability. When two modes are coupled and resonating with their phaseslocked together, the inertia of the mechanical domain, the tendency toremain at rest, increases and thus its fluctuation is reduced at thesame energy level of noise.

In an implementation, a resonator comprises a silicon microcantileverwhich is spanned to a firm substrate by a small polymer component. In animplementation, the dimensions of the structural components are chosento produce the desired 1:2 ratio between the second and third modalfrequencies: the length (L), width (b), and thickness (h) of the siliconmicrocantilever (subscript 1) and polymer coupling (subscript 2) areL₁=500 μm, b₁=100 μm, h₁=2 μm and L₂=40 μm, b₂=12 μm, h₂=3 μm. These,and other dimensions provided herein, are not intended to be limitingand it is contemplated that any appropriate dimensions may be useddepending on the implementation.

The thermomechanical response measured by a LDV shows that the firstthree linearized mode frequencies are f₁≅42 kHz, f₂≅107 kHz and f₃≅214kHz, and the second and third modal frequency values satisfy the 1:2relation of commensurability between the second and third modes of thesystem. The strong geometric nonlinearity in the heterogeneousnon-prismatic design combined with the 1:2 ratio between the modalfrequencies triggers the IR in the dynamic response. This implies thatthe second and third modal responses can be internally coupled if thesystem is driven hard enough into the nonlinear regime. Thus, theresponses in the second and third modes are monitored when one of thesemodes was externally driven by applying a single-frequency excitationaround the mode frequency.

FIGS. 10 and 11 are illustrations of the comparison of the harmoniccontent in fast Fourier transform-based (FFT-based) spectral responsesbetween the cases without (left column) and with (right column) IR showsthat IR acts as a mechanism that amplifies the undriven IRM by tunnelingthe energy from the ERM. The abbreviations LM and HM are used to denotethe lower-frequency (i.e., second) mode and higher-frequency (i.e.,third) mode, and ERM and IRM are used to denote the externally resonated(i.e., directly driven) mode and internally resonated mode. LME (LMexcitation) and HME (HM excitation) denote which mode is externallyexcited.

For LME with f_(drive)=106.04 kHz as shown in FIG. 10 , the amplitude ofIRM is increased from 1.34 nm to 36.35 nm as the IR is triggered, whilethe amplitude of ERM is increase from 30.17 nm to 242.5 nm. For HME withf_(drive)=258.28 kHz as shown in FIG. 13 , the amplitude of IRM isincreased from 1.35 nm to 825.8 nm, while the amplitude of ERM isincrease from 7.1 nm to 99.78 nm. Note that the undriven IRM is a higheroscillation amplitude than the directly driven ERM for HME.

While the existence of the sub-harmonics and/or super-harmonics in anonlinear dynamic response is not an uncommon phenomenon, the IRsubstantially enhances the amplitudes of those harmonics due to a strongintermodal energy transfer between the engaged modes. From the FFTresponses shown in FIGS. 10 and 11 , the amplitudes of these harmonicsare compared between the cases in which the IR is triggered (rightcolumn) and is not triggered (left column). As the IR is activated onlyif the input energy is higher than a critical value (see FIG. 14 ), thedrive voltage amplitude was tuned to either enter or escape the range ofIR.

FIGS. 12-14 show the experimental characterization of the nonlinear IRfrequency responses when the LM (left column) and HM (right column) areexternally driven. In FIG. 12 , the frequency responses of the ERM aredepicted during the upward (in circle) and downward (in asterisk)frequency sweeps at three different levels of the excitation.Experimental characterization of 1:2 and 2:1 IR when the LM (in leftcolumn) and HM (in right column) are driven. FIG. 12 shows amplitudes ofERM as a function of the driving frequency show the signature M-shape IRcurves at three different excitation amplitudes. The amplitudes duringthe upward and downward frequency sweep are shown in circle andasterisk, respectively. As the excitation amplitude increases, the IRactivation range expands and hysteresis manifests. FIG. 13 showsamplitudes of the ERM (at f_(drive)) and IRM (f_(IRM)=2f_(drive)) or(f_(IRM)=½ f_(drive)) shows coexistence of the two modes in the systemwhen IR is activated. The IR activation range is different depending onthe sweeping direction, which results in the hysteresis andjump-phenomena, marked by black arrows. FIG. 14 shows steady stateamplitudes of the ERM and IRM as a function of the driving voltages showthat there is a threshold energy for the onset of IR. It is clearlyshown that the external energy pumped to the ERM is transferred to theIRM once the IRM is activated. The energy transfer from ERM to IRM leadsto the amplitude saturation phenomenon in 2:1 IR.

The results demonstrate the typical M-shaped 1:2 IR response curves. Thehigher energy input to the system drives the system further into thenonlinear regime and expands the IR activation range. Eventually,hysteresis phenomena manifests because multiple stable branchesco-exist. FIG. 13 shows the amplitude the ERM and IRM with respect tothe driving frequency at the highest excitation voltage. As the drivingfrequency approaches the mode frequency from a low frequency, the ERMamplitude gradually increases. When the energy level of ERM surpasses acritical value, the IR mechanism is activated in the system and theamplitudes of both ERM and IRM are amplified. This intermodal nonlinearinteraction results in the vigorous energy exchange between the engagedmodes until the drop-jump phenomenon happens. These IR activation rangesare different depending on the sweeping direction, which results inhysteresis in both 1:2 and 2:1 IR responses. The hysteresis range iswider in 2:1 IR and one extra transition to an upper branch exists rightbefore the drop-down transition.

FIG. 14 shows the ERM and IRM amplitudes with respect to the excitationlevel at a fixed driving frequency (f_(drive)=107.5 kHz in the LME andf_(drive)=214.2 kHz in the HME). When the LM is driven at low voltagesless than 10V, the ERM amplitude increases linearly with the forcinglevel and the IRM amplitude is nearly zero. Due to the intrinsicgeometric nonlinearity of the system, there exists a higher (second)harmonic even when the IR is not triggered. Once IR is activated ataround 10V of the driving voltage, a sudden jump in the IRM amplitudeoccurs while energy continuously transfers from the directly driven ERMto the un-driven IRM. When the HM is externally driven, the so-calledamplitude saturation phenomenon occurs beyond a threshold forcing levelaround 5V, where the extra energy applied to the ERM is channeleddirectly into the un-driven IRM, and ERM amplitude keeps constant.Comparing 1:2 IR and 2:1 IR, one can conclude that the intermodal energytransfer from IRM to ERM is more vigorous and effective in the case ofthe HME as the amplitude of the IRM exceeds that of the ERM by an order.

An analytical model is provided based on the energy method to furtherunderstand the underlying dynamics in the nonlinear 1:2 and 2:1 IRsystems. The analytical results provide more detailed knowledge of thecomplex IR dynamics and the modal energy transfer. The patterns of thenonlinear resonances in IR systems drastically change depending on thetype of nonlinear couplings (i.e., quadratic or cubic), couplingstrength, internal detuning parameter, and forcing level. Thus, studyingthe effective parameters responsible for the unique resonance behaviorsis essential to explore IR in practical systems with the desiredresonance features.

To get the analytical model, first define the transverse displacement ofa beam of a resonator in which both the LM and HM are excited by a baseexcitation. When the base excitation frequency (Ω) is close to the LMfrequency (i.e., Ω=ω₁+ησ₂ where η is a small-scale parameter and σ₂ isan external frequency detuning parameter), the LM is harmonically drivenat the excitation frequency of Ω and the HM is internally resonated atthe frequency of 2Ω. Similarly, for the case of HME, the HM isexternally excited at Ω while the LM is internally resonated at ½Ω.Also, impose the internal frequency mismatch from the exact 1:2 ratiobetween the LM and HM frequencies to account for the potential deviationfrom the intended design in the wake of the fabrication errors andparameter randomness. In this regard, the relationship between the modalfrequencies is expressed with the equation ω₂=2ω₁+ησ₁ where σ₁ is aninternal frequency detuning parameter. Using the transverse displacementof a beam based upon these settings, the averaged Lagrangian andLagrange's equation are obtained to eventually deduce a set of leadingorder nonlinear equations governing the modal amplitudes. Theclamped-clamped structure inherently provides a nonlinear couplingbetween mode via mid-plane stretching.

The leading-order governing equations show that each LM or HM itself ismodeled as a linear harmonic oscillator with quadratic nonlinearcoupling originating from the axial strain (ϵ_(xx)). The axialstretching brings about the cubic coupling terms between the modalamplitudes of A₁ and A₂ (e.g., A₁ ³, A₂ ³, A₁ ²A₂, A₁A₂ ²) in the strainenergy, but only the term of A₁ ²A₂ remains as the only effectivenonlinear term in the time-averaged Lagrangian equation. Solving theseequations under the steady-state condition, the resulting dynamicbehaviors are analytically characterized under various sets of systemparameters to suggest the strategies to tailor the complex IR dynamics.

The nonlinear coupling terms are generated by the pure geometric(stretching) effect and, thus, determined by the geometric parametersand linear mode shapes of the engaged LM and HM. Therefore, one candesign the 1:2 IR systems with the targeted resonance behaviors bytailoring the geometric parameters. To suggest the design parametersthat can effectively integrate IR in micromechanical resonators andnanomechanical resonators, the effect of mode shapes is investigated byconsidering two sets of symmetric and asymmetric mode shapes. Thesesymmetric and asymmetric mode shapes are expressed by families of trialfunctions

${w_{n}(x)} = {{{\sin\left( {\frac{n\;\pi}{L}x} \right)}\mspace{14mu}{and}\mspace{14mu}{w_{n}(x)}} = {\sin\left( {\frac{n\;\pi}{L}x^{2}} \right)}}$for n=1, 2, 3 as depicted in FIGS. 15 and 16 , respectively. FIGS. 15and 16 respectively show that first three flexural mode shapes with asymmetrical configuration

${w_{n}(x)} = {\sin\left( {\frac{n\;\pi}{L}x} \right)}$and asymmetrical configuration with

${w_{n}(x)} = {\sin\left( {\frac{n\;\pi}{L}x^{2}} \right)}$for n=1, 2, 3.

The coupling coefficients are shown in Table I where other systemparameters are set to be same. Nonlinear coefficients in 1:2 IR systemswith symmetrical and asymmetrical flexural modes. Geometric parametersother than the mode shapes are set to be constant as ρ=1,

${\left( {\frac{\upsilon\;\mu}{1 - {2\upsilon}} + \mu} \right) = 1},$L=1, h=0.01.

TABLE I Beam with symmetrical Beam with asymmetrical mode shapes modeshapes Flexural mode numbers Flexural mode numbers 1^(st)-2^(nd)1^(st)-3^(rd) 2^(nd)-3^(rd) 1^(st)-2^(nd) 1^(st)-3^(rd) 2^(nd)-3^(rd)|α₁| 0 2.23 3.19 5.65 10.95 18.84 |α₂| 0 0.56 0.80 1.29 2.42 4.54

The results summarized in Table I suggest two noticeable facts. Firstly,the asymmetric mode shapes provide stronger intermodal coupling betweenany of the three modes compared to the symmetric mode shapes. Note thatin a prismatic beam with identical boundary conditions at both ends, itis not only difficult to achieve the integer n:m ratio between the modalfrequencies but also the coupling is weaker than the structures withasymmetric modes. Secondly, the strongest coupling occurs between thesecond and third modes amongst the lowest three flexural modes that arerelatively readily achievable in practice. These two attributes confirmthe validity of the mechanical resonator design in this study where 1:2ratio is implemented between the second and third modes in aheterogeneous non-prismatic beam.

Thus, according to some implementations, a geometrically nonlinearnon-prismatic IR system comprises a silicon microcantilever and polymercoupling that incorporates a 1:2 ratio between its second and thirdmodal frequencies. The commensurate relationship between the modescombined with the midplane stretching in the nonlinear system realizesthe IR dynamics with strong modal coupling. An analytical model is usedfor the quadratic IR systems based on the energy method for bothscenarios when the lower and higher modes are externally driven. Usingthis model, the characteristic behaviors of IR responses may be studiedwhile the effective parameters are varied over a range. The mechanism ofmodal energy transfer may be investigated at different values ofinternal detuning and nonlinear coefficients. The analytical model isable to provide a valuable insight about the IR mechanism and suggestdesign strategies to implement IR in a clamped-clamped beam structure:(i) the mid-plane stretching due to the constrained boundary conditionsprovides the nonlinear (quadratic) coupling mechanism between twoflexural modes, which is more dominant than the cubic geometricnonlinearity due to stretching of its own mode, (ii) the higher couplingrenders the wider IR dynamic range with a lower activation threshold,(iii) the mode shapes of the engaged modes determine the couplingstrength, and (iv) coupling second and third flexural modes in anasymmetric structure is a practically effective method to escalate theIR.

Targeting the desired IR response strongly relies on the accurateallocation of system parameters, such that small perturbations in theparameters can drastically alter the activation of IR, nonlinearresonances and bifurcation points. IR can be integrated in aclamped-clamped beam structure by modifying the geometric parameters tosatisfy the IR conditions. Even though the experimental demonstration isperformed in a non-prismatic beam with two materials (silicon andpolymer), a silicon beam with varying dimension (e.g., a stepped-beam, atapered-beam) can be similarly employed. A clamped-clamped beamstructure, that is most commonly used in MEMS/NEMS applications,provides a practical platform to take benefits from the dynamiccharacteristics originating from TR. The strategies suggested herein canbe readily extended to 2-dimensional plate structures as well.

MEMS/NEMS are great platforms to practically implement the IR dynamicsdue to their flexibility in design and fabrication. Besides, thefabrication randomness can be fairly easily overcome by frequencytunability of micro-resonators and nano-resonators (e.g., applyingtension through a gate DC voltage, changing the temperature).

Conventional microfabrication techniques may be used to producemicrocantilever patterns in the device layer of a silicon-on-insulatorwafer. Polyimide beams may be placed and delineated onto the pre-definedsilicon beams by blanket transferring and ensuing patterning, followedby deep reactive-ion etching (DRIE) of the silicon handle substrate toreveal the entire freestanding heterogeneous microstructure.

Advantages of the embodiments described or otherwise contemplated hereininclude: (1) the stepped-beam structure provides ample flexibility toadjust modal frequencies into n:m ratio during the design process, and(2) the thin-layer structure provides frequency tunability bycontrolling the mid-plane stretching effect with the applied DC bias andthus, the frequency mismatch from n:m ratio due to the fabrication errorcan be easily compensated.

According to some aspects, IR can be triggered and controlled readily inthe MEMS oscillator. Moreover, its frequency stability is improved morethan 30 times by this IR mechanism.

Regarding industry applications, MEMS oscillators have replaced theconventional mechanical oscillators (e.g., quartz oscillator) due to itsadvantage of high integrability with auxiliary electrical components.Thus, the low phase noise achieved by the embodiments can improve theMEMS devices applied for PNT (positioning, navigation, and timing).Moreover, the stable frequency operation is essential to improve thesensitivity of resonator-based MEMS sensors and thus the embodiments canbe broadly applied to sensing technology.

In an implementation, a device comprises: a non-prismatic resonatorconfigured to implement internal resonance (IR) to provide flexibilityto adjust modal frequencies into a n:m ratio, wherein n and m areintegers; and a plurality of electrodes configured to provide voltage tothe resonator.

Implementations may include some or all of the following features. Theresonator is one of a micromechanical resonator or a nanomechanicalresonator. The resonator is a microelectromechanical systems (MEMS)resonator. The MEMS resonator is an oscillator. A mistuning between twoflexural modes of the MEMS resonator can be precisely controlled bytuning a DC bias and, through strong coupling between the two flexuralmodes, a broader range of frequency stabilization is achieved byinternal resonance (IR). The resonator comprises at least one of astepped-beam structure that is a thin-layer structure, or aclamped-clamped beam structure. The resonator is configured to implementthe IR to adjust the modal frequencies into a n:m ratio, wherein m and nare different integers, each greater than one. The resonator isconfigured that when driven by a single-frequency actuation andconditions for IR are satisfied, two engaged modes are simultaneouslyresonated with their phases locked together through an intermodal energytransfer. Through DC bias tuning strategy, a 1:2 commensurate relationis achieved between second and third flexural modes. The resonatorcomprises a non-prismatic beam. The device further comprises a steppedclamped-clamped silicon microbeam. The stepped clamped-clamped siliconmicrobeam is fabricated by a MEMS fabrication flow to implement an IRmechanism.

In an implementation, a system comprises a microcantilever thatincorporates a 1:2 ratio between second and third modal frequencies ofthe microcantilever.

Implementations may include some or all of the following features. Themicrocantilever comprises silicon. The system is a geometricallynonlinear non-prismatic internal resonance (IR) system. A commensuraterelationship between modes of the cantilever combined with midplanestretching in the system realizes internal resonance (IR) with strongmodal coupling. The microcantilever comprises a stepped-beam structureconfigured to provide flexibility to adjust modal frequencies into a n:mratio, wherein n and m are integers. The microcantilever is configuredto adjust the modal frequencies into a n:m ratio, wherein m and n aredifferent integers, each greater than one. The microcantilever comprisesa thin-layer structure configured to provide frequency tunability bycontrolling a mid-plane stretching effect with an applied DC bias. Thethin-layer structure is configured to compensate for a frequencymismatch from a n:m ratio due to a fabrication error.

It should be understood that the various techniques described herein maybe implemented in connection with hardware components or softwarecomponents or, where appropriate, with a combination of both.Illustrative types of hardware components that can be used includeField-Programmable Gate Arrays (FPGAs), Application-specific IntegratedCircuits (ASICs), Application-specific Standard Products (ASSPs),System-on-a-chip systems (SOCs), Complex Programmable Logic Devices(CPLDs), etc. The methods and apparatus of the presently disclosedsubject matter, or certain aspects or portions thereof, may take theform of program code (i.e., instructions) embodied in tangible media,such as floppy diskettes, CD-ROMs, hard drives, or any othermachine-readable storage medium where, when the program code is loadedinto and executed by a machine, such as a computer, the machine becomesan apparatus for practicing the presently disclosed subject matter.

Although exemplary implementations may refer to utilizing aspects of thepresently disclosed subject matter in the context of one or morestand-alone computer systems, the subject matter is not so limited, butrather may be implemented in connection with any computing environment,such as a network or distributed computing environment. Still further,aspects of the presently disclosed subject matter may be implemented inor across a plurality of processing chips or devices, and storage maysimilarly be effected across a plurality of devices. Such devices mightinclude personal computers, network servers, and handheld devices, forexample.

Although the subject matter has been described in language specific tostructural features and/or methodological acts, it is to be understoodthat the subject matter defined in the appended claims is notnecessarily limited to the specific features or acts described above.Rather, the specific features and acts described above are disclosed asexample forms of implementing the claims.

What is claimed:
 1. A device comprising: a non-prismatic resonatorconfigured to implement internal resonance (IR) to provide flexibilityto adjust modal frequencies into a n:m ratio, wherein n and m areintegers; and a plurality of electrodes configured to provide voltage tothe resonator, wherein the non-prismatic resonator is a clamped-clampedsilicon structure with one or more steps.
 2. The device of claim 1,wherein the resonator is one of a micromechanical resonator or ananomechanical resonator.
 3. The device of claim 1, wherein theresonator is a microelectromechanical systems (MEMS) resonator.
 4. Thedevice of claim 3, wherein the MEMS resonator is an oscillator.
 5. Thedevice of claim 3, wherein a mistuning between two flexural modes of theMEMS resonator can be precisely controlled by tuning a DC bias and,through strong coupling between the two flexural modes, a broader rangeof frequency stabilization is achieved by internal resonance (IR). 6.The device of claim 1, wherein the resonator comprises at least one of astepped-beam structure that is a thin-layer structure, or aclamped-clamped beam structure.
 7. The device of claim 1, wherein theresonator is configured to implement the IR to adjust the modalfrequencies into a n:m ratio, wherein m and n are different integers,each equal or greater than one.
 8. The device of claim 1, wherein theresonator is configured that when driven by a single-frequency actuationand conditions for IR are satisfied, two engaged modes aresimultaneously resonated with their phases locked together through anintermodal energy transfer.
 9. The device of claim 1, wherein through DCbias tuning strategy, a 1:2 commensurate relation is achieved betweensecond and third flexural modes.
 10. The device of claim 1, wherein theresonator comprises a non-prismatic beam.
 11. The device of claim 1,wherein the stepped clamped-clamped silicon microbeam is fabricated by aMEMS fabrication flow to implement an IR mechanism.
 12. A systemcomprising: a micro-resonator that incorporates a 1:2 ratio betweensecond and third modal frequencies of the micro-resonator, wherein themicro-resonator comprises a structure with one or more steps configuredto provide flexibility to adjust modal frequencies into a n:m ratio,wherein n and m are integers.
 13. The system of claim 12, wherein themicro-resonator comprises silicon.
 14. The system of claim 12, whereinthe system is a geometrically nonlinear non-prismatic internal resonance(IR) system.
 15. A system comprising: a micro-resonator thatincorporates a 1:2 ratio between second and third modal frequencies ofthe microcantilever, wherein a commensurate relationship between modesof the cantilever combined with midplane stretching in the systemrealizes internal resonance (IR) with strong modal coupling.
 16. Thesystem of claim 12, wherein the micro-resonator is configured to adjustthe modal frequencies into a n:m ratio, wherein m and n are differentintegers, each greater than one.
 17. A system comprising: amicro-resonator that incorporates a 1:2 ratio between second and thirdmodal frequencies of the micro-resonator, wherein the micro-resonatorcomprises a thin-layer structure configured to provide frequencytunability by controlling a mid-plane stretching effect with an appliedDC bias.
 18. The system of claim 17, wherein the thin-layer structure isconfigured to compensate for a frequency mismatch from a n:m ratio dueto a fabrication error.